C₃×#27162404 #49181765

order	37926
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃×#27162404

Splittings

Fitting, socle	C₃×C₇²×C₄₃ ⋊ C₂×C₃
derived, Hall	C₇²×C₄₃ ⋊ C₂×C₃²
Hall	C₃×#20263479 ⋊ C₂
Hall	C₇⋊D₃₀₁ ⋊ C₃²
Sylow	C₇² ⋊ C₃×D₄₃⋊C₃
Sylow	C₄₃ ⋊ C₃×#64424

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#27162404
commutator, cosocle, hypocenter	C₇²×C₄₃	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₄₃	C₂×C₃
LCS	G ▷ C₇²×C₄₃
UCS	C₁ ◁ C₃

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 6321

A minimal faithful permutation representation — 3 generators on 60 points:

(5,8,9,6,7,10)(12,15,16,13,14,17)(19,23,24,20,21,25)(22,31,32,26,27,41)(28,44,45,34,35,53)(29,43,39,37,33,30)(36,49,51,40,42,54)(38,55,56,47,46,50)(48,58,59,57,52,60);(1,2,3)(11,12,14,17,16,15,13)(18,19,22,28,36,39,24,23,29,38,48,56,32,43,31,42,52,59,45,54,55,44,35,46,51,53,60,58,49,27,33,41,50,57,47,37,21,25,30,40,34,26,20);(4,5,7,10,9,8,6)

Conjugacy classes & character table

1071 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37926, 338);

C3×#27162404 #49181765